Galois group: 7T5

• Label: 7T5
• Degree: $7$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $\GL(3,2)$

Group action invariants

Degree $n$:		$7$
Transitive number $t$:		$5$
Group:		$\GL(3,2)$
CHM label:		$L(7) = L(3,2)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2)(3,6), (1,2,3,4,5,6,7)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{7}$	$1$	$1$	$0$	$()$
2A	$2^{2},1^{3}$	$21$	$2$	$2$	$(2,7)(3,4)$
3A	$3^{2},1$	$56$	$3$	$4$	$(1,7,6)(2,5,3)$
4A	$4,2,1$	$42$	$4$	$4$	$(1,5)(2,4,7,3)$
7A1	$7$	$24$	$7$	$6$	$(1,6,7,5,2,4,3)$
7A-1	$7$	$24$	$7$	$6$	$(1,4,5,6,3,2,7)$

Malle's constant $a(G)$:     $1/2$

Group invariants

Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		168.42
Character table:

		1A	2A	3A	4A	7A1	7A-1
	Size	1	21	56	42	24	24
	2 P	1A	1A	3A	2A	7A1	7A-1
	3 P	1A	2A	1A	4A	7A-1	7A1
	7 P	1A	2A	3A	4A	1A	1A
	Type
168.42.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
168.42.3a1	C	$3$	$-1$	$0$	$1$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.42.3a2	C	$3$	$-1$	$0$	$1$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.42.6a	R	$6$	$2$	$0$	$0$	$-1$	$-1$
168.42.7a	R	$7$	$-1$	$1$	$-1$	$0$	$0$
168.42.8a	R	$8$	$0$	$-1$	$0$	$1$	$1$

Additional information

This is the lowest degree transitive permutation group which admits a Gassmann triple. As a result, number fields with this Galois group come in arithmetically equivalent pairs.